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lc_circuit [2015/06/07 11:06] (current)
admin created
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 +====== LC Oscillator Time Domain Analysis ======
  
 +{{:​lc_circuit.png?​400|}}
 +
 +----
 +===== Nodal Equation =====
 +
 +\[i_C+i_L=0\]
 +
 +\[C\frac{dv_{OUT}}{dt}+\frac{1}{L}\int v_{out}=0\]
 +
 +\[\frac{d^2v_{OUT}}{dt^2}+\frac{1}{LC}v_{OUT}=0\]
 +
 +----
 +===== Solution using method of homogeneous and particular solutions =====
 +
 +First, notice that the nodal equation is a second order homogeneous equation.
 +Therefore, we will not need to find a particular solution.
 + 
 +As usual with problems like these, guess that the solution is $e^{st}$.
 +
 +Plugging in $v_{OUT=}e^{st}$ into the nodal equation yields:
 +
 +\[e^{st}\left ( s^2 + \frac{1}{LC} \right )\]
 +
 +Solving the above equation for s gives:
 +
 +\[s=\pm \sqrt{\frac{-1}{LC}}\]
 +
 +\[s=\pm j\sqrt{\frac{1}{LC}}\]
 +
 +So..
 +
 +\[v_{out}=e^{j\sqrt{\frac{1}{LC}}t}\]
 +
 +and
 +
 +\[v_{out}=e^{-j\sqrt{\frac{1}{LC}}t}\]
 +
 +There are two properties of homogeneous differential equations that are useful. ​ A constant times the solution is still a solution, and the sum of solutions is still a solution.
 +
 +Using those two properties we can write:
 +
 +\[v_{out}=Ae^{j\sqrt{\frac{1}{LC}}t}+Ae^{-j\sqrt{\frac{1}{LC}}t}\]
 +
 +Any good math student will recognize that is just,
 +
 +\[v_{out}=A'​cos \left [t\sqrt{\frac{1}{LC}} \right ]\]
 +
 +At $t=0$, $v_{OUT}=v_{IN}$.
 +
 +Finally we get,
 +
 +\[v_{out}=v_{IN}cos \left [t\sqrt{\frac{1}{LC}} \right ]\]
 +
 +----
 +
 +===== Example =====
 +
 +Let's try to make a 1khz oscillator. ​
 +
 +Remember that $w=2\pi f$.
 +
 +\[f=\frac{1}{2 \pi\sqrt{LC}}\]
 +
 +\[1000=\frac{1}{2 \pi\sqrt{LC}}\]
 +
 +There are many answers that work...Here is one combination of L and C that produces a 1khz oscillator. ​
 +
 +{{:​lc_simulation.jpg?​800|}}
lc_circuit.txt ยท Last modified: 2015/06/07 11:06 by admin